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Varieties of Associative Rings Containing a Finite Ring that is Nonrepresentable by a Matrix Ring Over a Commutative Ring

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In this paper, we give examples of infinite series of finite rings B (m) v , where m ≥ 2, 0 ≤ v ≤ p−1, and p is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety var B (m) v , where m = 2 or m − 1 = (p − 1)k, k ≥ 1, and p ≥ 3 or p = 2 and m ≥ 3, 0 ≤ v < p, and p is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order p.

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Authors and Affiliations

  1. Institute of Mathematics, Mongolian State University, Ulaanbaatar, Mongolia

    A. Mekei

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  1. A. Mekei
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A. Mekei is deceased.

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 2, pp. 187–206, 2014.

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Mekei, A. Varieties of Associative Rings Containing a Finite Ring that is Nonrepresentable by a Matrix Ring Over a Commutative Ring. J Math Sci 213, 254–267 (2016). https://doi.org/10.1007/s10958-016-2714-4

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